where \( Z(\Delta )\)  is a wave function renormalisation factor related to the form factors \(W_{R,k}(\Delta )\)  and \(W_{Ricci,k}(\Delta )\) ,
which have been computed in various approximations in [1]}, [2]}, [3]}, [4]}, [5]}, [6]} and the physical implications for scattering amplitudes have been discussed in [7]}, [8]}, [9]}, [10]}. (In principle there can be another independent wave function renormalisation related to the scalar degree of freedom that is introduced in theories such as \(f(R)\)  gravity. For simplicity, we discuss the case where there is only one wave function renormalisation which implies a linear relation between \(W_{R,k}(\Delta )\)  and \(W_{Ricci,k}(\Delta )\) ). There are two cases, either \( Z\)  introduces new poles into the propagator, or it does not. In the latter case, we can remove \(Z\)  by a reparameterisation since it must be an entire function, and thus it is just a momentum-dependent wave function renormalisation.
In this case, we will find the same physics as in the minimal essential scheme, namely, although the field redefinition would modify the vertices of the theory, the propagator would return to the minimal form (REF ). The case where \(Z\)  is not an entire function corresponds to a universality class not accessible to the minimal essential scheme for pure gravity. In particular, it would include particles other than the massless graviton.
Thus, on one hand, the minimal essential scheme for quantum gravity, like its counterpart of scalar field theories [11]}, does put a restriction on what physics we can access by following the corresponding RG flow. On the other hand, this is a feature of the scheme, and not a bug, since the restricted theory space has a physical meaning, describing the interactions of gravitons which are fluctuation around a flat spacetime.
Moreover, there is no reason why these fluctuations can not be strongly interacting, in particular \(\tilde{G}\)  can become of order unity.
